Struggling with Integrals? Don't Panic, Here's Some Tips!

[1MileCrash]

Homework Statement

$\int sin^{2}x dx$

Homework Equations

The Attempt at a Solution

I don't see why. Clearly it is designed for me to shoot myself before finishing.

u = (sinx)^2
du = 2sinxcosx dx, magically equal to sin2x dx
v = x
dv = dx

(-xcos2x/2) + (sin2x/4)

=

(-2xcos2x+sin2x)/(4) + C = wrong

 

[SammyS]

Since you just reviewed the double angle formula for sin, what's the double angle formula for cos?

cos(2x) = ? = ? = ?

Yes, there are three forms. One will work quite nicely here.

 

[1MileCrash]

Firstly, let me apologize for my attitude. Secondly, thanks for the tip, I'll work it again and post back.

 

[SammyS]

At times, getting frustrated can lead to good things.

 

[1MileCrash]

$\int sin^{2}x dx$

$u = sin^{2}x$
$du = sin2x dx$
$v = x$
$dv = dx$

$xsin^{2}x - \int xsin2x dx$

$u = x$
$du = dx$
$v = -\frac{cos2x}{2}$
$dv = sin2x dx$

$xsin^{2}x - [-\frac{xcos2x}{2} - \int -\frac{cos2x}{2} dx ]$
$xsin^{2}x - [-\frac{xcos2x}{2} + \frac{sin2x}{4}]$

$xsin^{2}x + \frac{xcos2x}{2} - \frac{sin2x}{4}$

$\frac{2xsin^{2}x + xcos2x}{2} - \frac{sin2x}{4}$


$\frac{2xsin^{2}x + x(cos^{2}x - sin^{2}x)}{2} - \frac{sin2x}{4}$


$\frac{2xsin^{2}x + xcos^{2}x - xsin^{2}x}{2} - \frac{sin2x}{4}$


$\frac{xsin^{2}x + xcos^{2}x}{2} - \frac{sin2x}{4}$


$\frac{x(sin^{2}x + cos^{2}x)}{2} - \frac{sin2x}{4}$


$\frac{x}{2} - \frac{sin2x}{4}$

 

[Mark44]

There's a much shorter way of doing this, following Sammy's hint.

sin²(x) = 1/2 * (1 - cos(2x))

This will give you an integral that you can do without having to resort to integration by parts.

 

[SammyS]

Mark44's expression comes from solving the following for sin²(x) .

cos(2x) = 1 - 2 sin²(x) .

It's also true that

cos(2x) = cos²(x) - sin²(x)

cos(2x) = 2 cos²(x) - 1

 

[Ocasta]

Firstly, let me apologize for my attitude.

Welcome to the wonderful world of integrals! There are times when I want to kick small, defenseless animals because of this stuff. I usually listen to some thrash metal while studying math. I don't know why, but it helps!

Try to remember as many identities as you possibly can. It makes life a lot easier.